The spec for a cigarette lighter plug/socket is 10 amps. Some have been designed as "power" sockets for currents higher than that, but the pedestrian socket in most any vehicle isn't really up to powering a high current device like an inflator. Powering a phone or radar detector is pretty much all they're designed to handle. Of course the typical user doesn't know that, believing that plugs and sockets that fit together will work. So much for ubiquitous compatibility. It's my guess that this circuit popped as intended by design, with a fuse in place suited for the light gauge wire that's likely there.
This probably isn't directly useful advice since I have a different chassis, but the fuse panel for my accessory sockets is behind and to the left of the instrument cluster inside the dash. So I have to tilt up the instrument cluster to get at it, and it's separate from the fuse panel on the firewall left of the brake pedal. So throwing it out there that if your instrument cluster tilts/hinges for access, give a peek for a panel in there.
Mark B.
Albuquerque, NM